negative numbers but imaginary num-
bers in his Latin treatise Ars Magna(The
Great Art). But Cardano did not consider
the imaginary numbers as the real math-
ematical objects we do today. To him,
they were merely convenient “fiction” to
classify certain polynomial properties,
describing how their roots would behave
when he pretended they even hadroots.
Most agree that around 1777, Swiss
mathematician Leonhard Euler (1707–
1783) used “i” and “i” (negative i) for the
two different square roots of 1, thus
eliminating some of the problems associ-
ated with notation when putting polyno-
mials into categories. (He is also credited
with originating the notation abifor
complex numbers.) Much to the conster-
nation of many past and present mathe-
maticians, iand iwere called “imagi-
nary,” mainly because the number’s
function at the time of Euler was not
clearly understood. When German mathe-
matician, physicist, and astronomer
Johann Friedrich Carl Gauss (1777–1855)
used them for the geometric interpretation of complex numbers as points in a plane, the
usefulness of imaginary numbers became apparent. (For more information on Gauss,
Cardano, and Euler, see “History of Mathematics.”)
What are complex numbers?
Complex numbers have two parts: a “real” part (any real number) and an “imaginary”
part (any number with an iin it). The standard complex number format is “abi,” or
a real number plus an imaginary number. It is also often seen as xiybecause while
real numbers are viewed on a line, complex numbers are viewed graphically on an
Argand (or polar) coordinate system: The imaginary numbers make up the vertical (or
y) axis as iy,while the horizontal (or x) axis is occupied by real numbers. (For more
information about coordinate systems, see “Geometry and Trigonometry.”)
What is the polar formof a complex number?
The polar form of a complex number is equal to a real number expressed as an angle’s
cosine, and the imaginary number (i) times the same angle’s sine, with the angle 77
MATH BASICS
Eighteenth-century Swiss mathematician Leonhard
Euler, who published over 70 volumes on mathe-
matics in his lifetime, was one of the greatest con-
tributors to the discipline that ever lived. Euler
developed important concepts in such areas as
geometry, calculus, trigonometry, algebra, hydrody-
namics, and much more. He also created the con-
cept of the imaginary number that is the square root
of 1. Library of Congress.